52 



BIRKHOFF AND LANGER. 



Section I. 

 Definitions. 3 

 An array of elements of the form 



On «i2 fli.-i 



0,21 f'22 .... 

 «31 



Cl3„ 



Onl On2- 



in which the number of rows equals the number of columns is called a 

 square matrix, and is denoted either by (o i; ) or by A. Two such 

 matrices are said to be equal when, and only when, every element of 

 the one is equal to the correspondingly situated element of the other. 



The sum of two matrices, («»,-) and (&#), having the same number of 

 rows and columns, is by definition the matrix fan -f ft*,-), from which 

 it follows that matrix addition is both commutative and associative, 

 i.e., A + B = B + A, and A + (B + C) = (A + B)+C. 



The product of two n rowed matrices A and B is defined by the 

 identical equation 



M faid 



n 



-k=l 



aikbkj), 



on the basis of which it is easily verified that matrix multiplication is 

 both associative and distributive, i.e. 



A(BC) = (AB)C, 

 A(B + C) = ABA- AC. 



Differential Equations Containing a Parameter, and Boundary Value and Ex- 

 pansion Problems of Ordinary Linear Differential Equations, vol. 9 (190S), p. 

 219 and p. 373. 



Wilder, Expansion Problems of Ordinary Linear Differential Equations with 

 Auxiliary Conditions at More than Two Points, vol. 18 (1917), p. 415. 



Hopkins, Some Convergent Developments Associated with Irregular Boundary 

 Conditions, vol. 20 (1919), p. 245. 



Hurwitz, An Expansion Theorem for a System of Linear Differential Equa- 

 tions of the First Order (about to appear in vol. 22 (1921)). 



Langer, Developments Associated with a Boundary Problem not Linear in the 

 Parameter (about to appear in vol. 22 (1921)). 



3 For a more ample discussion of the theory of matrices see Bocher, M., In- 

 troduction to Higher Algebra. New York; The Macmillan Co., 1907. 



